2013 Issue 438

  1. Determine all triples of (not necessarily distinct) prome numbers $a,b,c$ such that \[a(a+1)+b(b+1)=c(c+1).\]
  2. In an isosceles right triangle $ABC$, right angle at vertex $A$, let $E$ be the midpoint of side $AC$. The line perpendicular to $BE$ through $A$ meets $BC$ at $D$. Prove that $AD=2ED$. 
  3. Find all positive integers $x,y$ and $t$ such that $t\leq6$ and the following equation is satisfied \[x^{2}+y^{2}-4x-2y-7t-2=0.\]
  4. Find the greatest integer not exceeding $A$, where \[A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{2013}}.\]
  5. Let $ABC$ be an acute triangle with altitudes $BE$ and $CF$. The line segments $FI$ and $EJ$ are perpendicular to $BC$ ($I,J$ belong to $BC$). Point $K,L$ on $AB$, $AC$ respectively such that $KI||AC$, $LJ||AB$. Prove that the lines $EI$, $FJ$ and $KL$ are concurrent. 
  6. Solve for $x$ \[\sqrt{13}\sqrt{2x^{2}-x^{4}}+9\sqrt{2x^{2}+x^{4}}=32.\]
  7. Given that $x,y$ satisfy the conditions \[x-y\geq0,\,x+y\geq0,\,\sqrt{\left(\frac{x+y}{2}\right)^{3}}+\sqrt{\left(\frac{x-y}{2}\right)^{3}}=27.\] Find the smallest possible value of $x$.
  8. In a triangle $ABC$, let $m_{a},l_{b},l_{c}$ and $p$ denote the median length from $A$, the lengths of the internal angle bisectors from $B,C$ and its semiperimeter. Prove that \[m_{a}+l_{b}+l_{c}\leq p\sqrt{3}.\]
  9. a) Let $a_{1},a_{2},\ldots,a_{n}$ be $n$ rational numbers. Prove that if $a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$ is an integer for all positive integers $m$, then $a_{1},a_{2},\ldots a_{n}$ are integer numbers.
    b) Does the conclusion above remain valid if one only assumes that $a_{1},a_{2},\ldots a_{n}$ are real numbers?. 
  10. Find all continuous funtions $f:\mathbb{R}\to\mathbb{R}$ such that the following identity is satisfied for all $x,y$ \[ f(x)+f(y)+2=2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right).\]
  11. Let $ABC$ be a triangle with side lengths $BC=a$, $AC=b$, $AB=c$ and median lengths $m_{a},m_{b},m_{c}$ drawn from vertex $A,B$ and $C$ respectively. Prove that for any point $M$ \begin{align*}\left(\frac{m_{b}}{b}+\frac{m_{c}}{c}\right)\frac{MA}{a}+\left(\frac{m_{c}}{c}+\frac{m_{a}}{a}\right)\frac{MB}{a}+\left(\frac{m_{a}}{a}+\frac{m_{b}}{b}\right)\frac{MC}{c} & \geq3,\\ \left(\frac{b}{m_{b}}+\frac{c}{m_{c}}\right)\frac{MA}{a}+\left(\frac{c}{m_{c}}+\frac{a}{m_{a}}\right)\frac{MB}{a}+\left(\frac{a}{m_{a}}+\frac{b}{m_{b}}\right)\frac{MC}{c} & \geq4.\end{align*}
  12. Let $ABC$ be an isosceles triangle at vertex $A$. A circle $\omega$ which touches the sides $AB,AC$ meets $BC$ at $K,L$. $AK$ meets $\omega$ at $M$. Let $P,Q$ be the reflection points of $K$ through points $B,C$ respectively. Let $O$ be the circumcenter of triangle $MPQ$. Prove that $M$, $O$ and the center of circle $\omega$ are collinear.




Mathematics & Youth: 2013 Issue 438
2013 Issue 438
Mathematics & Youth
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